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Research on the theories of viscosity solution, weak KAM, and application to the asymptotic analysis on Hamilton-Jacobi equations

Research Project

Project/Area Number 17KK0093
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research)

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionThe University of Tokyo (2018-2020)
Hiroshima University (2017)

Principal Investigator

Mitake Hiroyoshi  東京大学, 大学院数理科学研究科, 准教授 (90631979)

Project Period (FY) 2018 – 2020
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥14,300,000 (Direct Cost: ¥11,000,000、Indirect Cost: ¥3,300,000)
Keywordsハミルトン・ヤコビ方程式 / 均質化問題 / 長時間挙動 / 粘性解理論 / Aubry-Mather理論 / 弱KAM理論 / 収束率
Outline of Final Research Achievements

The main subject of my research is nonlinear Partial Differential Equations, and in particular I have made some special effort to the study on the Hamilton-Jacobi (HJ) equation. This equation is an important fundamental equation for various branches of science like classical mechanics, geometric optics, calculus of variations, optimal control and differential games. During the project period, I focused on problems related to various properties of viscosity solutions of Hamilton-Jacobi equations appearing in the context of classical mechanics and crystal growth. In particular, I have worked on the following topics: (a) Asymptotic analysis on Hamilton-Jacobi-Bellman equations (the large time behavior, homogenization), (b) Analysis on the birth-and-spread model equation appearing in the crystal growth, (c) Selection problems for the mean field game. I got several new and important results and published 9 (peer-reviewed) papers.

Academic Significance and Societal Importance of the Research Achievements

補助事業期間中に,最適確率制御問題に現れる退化粘性ハミルトン・ヤコビ方程式と呼ばれるクラスの方程式に適用できるよう,弱 KAM 理論の一般化に取り組んだ.従来の弱 KAM 理論は決定論的な力学系しか扱えないため,新しい道具立てを必要とした.この点を偏微分方程式論から見直すことで決定論及び確率論を 統一する一つの新しい枠組みを幾つか作ることに成功してきた.応用として,漸近解析(長時間挙動,均質化問題)ついて解決した.これらの成果は,偏微分方程式論における粘性解理論,弱KAM理論において,特に重要な学術的意義を持つと期待できる.

Report

(4 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (24 results)

All 2020 2019

All Int'l Joint Research (1 results) Journal Article (6 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 6 results) Presentation (16 results) (of which Int'l Joint Research: 7 results,  Invited: 16 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of California, Irvine(米国)2019

    • Related Report
      2020 Annual Research Report
  • [Journal Article] Generalized ergodic problems: existence and uniqueness structures of solutions2020

    • Author(s)
      W. Jing, H. Mitake, H. V. Tran
    • Journal Title

      J. Differential Equations

      Volume: 268 Issue: 6 Pages: 2886-2909

    • DOI

      10.1016/j.jde.2019.09.046

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Discrete schemes for time-fractional fully nonlinear evolution equations and their convergence2020

    • Author(s)
      Y. Giga, Q. Liu, H. Mitake
    • Journal Title

      Asymptot. Anal.

      Volume: accepted Issue: 1-2 Pages: 1-12

    • DOI

      10.3233/asy-191583

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The selection problem for some first-order stationary Mean-field games2020

    • Author(s)
      D. A. Gomes, H. Mitake, K. Terai,
    • Journal Title

      Networks & Heterogeneous Media

      Volume: 15 Issue: 4 Pages: 681-710

    • DOI

      10.3934/nhm.2020019

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A level set approach for multi-layered interface systems2020

    • Author(s)
      Mitake Hiroyoshi、Ninomiya Hirokazu、Todoroki Kenta
    • Journal Title

      Interfaces and Free Boundaries

      Volume: 22 Issue: 4 Pages: 383-400

    • DOI

      10.4171/ifb/444

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms2019

    • Author(s)
      Y. Giga, H. Mitake and H. V. Tran
    • Journal Title

      Discrete Contin. Dyn. Syst. Ser. B

      Volume: Online First Issue: 10 Pages: 3983-3999

    • DOI

      10.3934/dcdsb.2019228

    • NAID

      120006643619

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Rate of Convergence in Periodic Homogenization of Hamilton-Jacobi Equations: The Convex Setting2019

    • Author(s)
      H. Mitake, H. V. Tran, Y. Yu
    • Journal Title

      Archive for Rational Mechanics and Analysis

      Volume: 印刷中 Issue: 2 Pages: 901-934

    • DOI

      10.1007/s00205-019-01371-y

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model2020

    • Author(s)
      H. Mitake
    • Organizer
      2020 Seoul-Tokyo Conference
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 退化粘性Hamilton--Jacobi方程式の長時間挙動2020

    • Author(s)
      H. Mitake
    • Organizer
      2020年度「微分方程式の総合的研究」
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 凝結・分裂モデルに現れるハミルトン・ヤコビ方程式2020

    • Author(s)
      H. Mitake
    • Organizer
      第38回 九州における偏微分方程式研究集会
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 外力付き平均曲率流方程式の一般化ディリクレ問題について2020

    • Author(s)
      三竹 大寿
    • Organizer
      九州関数方程式セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 粘性ハミルトン・ヤコビ方程式に対する弱KAM理論とその応用2019

    • Author(s)
      三竹 大寿
    • Organizer
      第688回早稲田大学 応用解析研究会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 退化粘性ハミルトン・ヤコビ方程式とその解の漸近形について2019

    • Author(s)
      三竹 大寿
    • Organizer
      金沢解析セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] On approximation of time-fractional fully nonlinear equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      Recent Progress in Nonlinear Partial Differential Equations
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The large-time profile for Hamilton--Jacobi--Bellman equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      New trends in Hamilton-Jacobi: PDE, Control, Dynamical Systems and Geometry
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the generalized Dirichlet problem for graph mean curvature flow with driving force2019

    • Author(s)
      三竹 大寿
    • Organizer
      RIMS共同研究(公開型)「非線形偏微分方程式における定性的理論」
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On birth and spread type nonlinear partial differential equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      Geometric and Analysis
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 退化粘性ハミルトン・ヤコビ方程式の弱KAM理論2019

    • Author(s)
      三竹 大寿
    • Organizer
      第7回ハミルトン系とその周辺
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 退化粘性ハミルトン・ヤコビ方程式の解の長時間挙動2019

    • Author(s)
      三竹 大寿
    • Organizer
      京都大学NLPDEセミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Rate of convergence in periodic homogenization of Hamilton-Jacobi equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      第36回九州における偏微分方程式研究集会
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The large-time profile for Hamilton-Jacobi equations revisited2019

    • Author(s)
      三竹 大寿
    • Organizer
      確率論と幾何学
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The large-time profile for viscous Hamilton-Jacobi equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      Conference on PDEs, Dynamical Systems and Probability
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] The large-time profile for Hamilton--Jacobi--Bellman equations2019

    • Author(s)
      三竹 大寿
    • Organizer
      第11回 名古屋微分方程式研究集会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Funded Workshop] 界面運動,力学系に現れる漸近問題への粘性解的手法とその周辺2019

    • Related Report
      2019 Research-status Report

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Published: 2018-01-25   Modified: 2022-01-27  

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